The computational complexity of knot and link problems

Abstract

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, i.e., capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, UNKNOTTING PROBLEM is in NP. We also consider the problem, SPLITTING PROBLEM of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worst-case running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.

The computational complexity of basic decision problems in 3-dimensional topology

“…As a corollary of Theorem 3 (in case when M is a solid torus), we recover the main result of Hass, Lagarias and Pippenger [6].…”

“…We remark that, for the case of an irreducible 3-manifold, Theorem 4 can be derived from [13,Corollary 6.4] and [6,Lemma 6.1]. However, in general case, we apply essentially same arguments as those used in the proof of Theorem 3. Recall that the complexity classes NP, coNP, PSPACE are known to satisfy NP ∪ coNP ⊆ PSPACE and PSPACE = coPSPACE, see [17].…”

“…We refer the reader for more details to [6,8,11,13] and give here a brief outline in order to discuss necessary adjustments and introduce the notation and definitions that we need in the sequel.…”

“…Proof To prove this, we essentially repeat the proof of part (1) of Lemma 6.1 [6]. First we rewrite SMI(M ) in the matrix form Ax ≤ b, where x = (x 1 , .…”

“…Since a nontrivial -normal 2-sphere contains either a normal quadrangle, when = 1, see Lemma 9, or an -normal disk, when ≥ 2, it follows that an S-reduction necessarily decreases the number of tetrahedra in M i, i and so the total number of S-reductions is again at most N T ( ). Therefore, in view of inequality (6) and Lemma 11, if M ∈ K, then there is a certificate τ , using which, our reduction process never stops and destroys all the tetrahedra of F ( ) so that the final 3-manifold M and are undefined (here we use the notation of Theorem 6). As in the proof of Theorem 6, making use of the simplicial homology of the 2-complex M (2), we can find the numbers k 0 + k h , k 2 , k 3 in linear time of σ ( ).…”